The Dirichlet problem for the Monge-Ampere equation in convex (but not strictly convex) domains

Abstract

It is well-known that the Dirichlet problem for the Monge-Amp`ere equation $det D^2 u = mu$ in a bounded strictly convex domain $Omega$ in $mathbb{R}^n$ has a weak solution (in the sense of Aleksandrov) for any finite Borel measure $mu$ on $Omega$ and for any continuous boundary data. We consider the Dirichlet problem when $Omega$ is only assumed to be convex, and give a necessary and sufficient condition on the boundary data for solvability.

Keywords

• Aleksandrov solutions
• Perron method
• viscosity solutions.